Due to important contributions from several people it is now known that a compact set \(E\subset \mathbb {R}^n\)  with \(\mathcal {H}^{n-1}(E)>0\)  is removable for Lipschitz harmonic functions if and only if \(E\)  is purely \((n-1)\) –unrectifiable, that is, \(E\)  intersects every \(\mathcal {C}^1\)  hypersurface in a set of vanishing \((n-1)\) –dimensional Hausdorff measure. One of the key ingredients in the proof of the “only if” direction is the \(L_2(\Gamma )\) –boundedness of \(T^{n-1}\)  for Lipschitz graphs of codimension 1. The harder “if” direction was proved by David and Mattila [1]} (for \(n=2\) ), and Nazarov, Tolsa and Volberg [2]}, [3]} for \(n\ge 3\) . We also mention that the \(L_2\) –boundedness of the Cauchy transform/1–dimensional Riesz transform was the key tool in geometrically characterizing removable sets for bounded analytic functions, see [4]}, [5]} for the long and interesting history of this problem.
